Here is the diagram for your car. Always start with the right diagrams because they have the location, color and size of the wires to help in your repairs.
http://www.mymopar.com/downloads/1968/68BelvedereGTXSatelliteRoadRunneFull.pdfNow for your questions.
I have bypassed the factory ammeter wires on a few occasions like you asked. Like Rapidrobert said it's better as a proactive approach. THE REAL PROBLEM WITH ADDING WIRES is in the junction in the main underdash harness. You can run a larger wire from the alternator to the back of the ammeter and then another wire from the ammeter to the fusible link, then battery. Preferably with a better connection than the bulkhead had from the factory.
That will augment the harness and reduce the load on the factory wires but there is a junction inside the harness that feeds the major circuits. You must connect the extra wires and the factory wires to the back of the ammeter. If not it you will not have power to any of the cars circuits including the car's ignition switch, headlights, fuse block etc. If the wiring is damaged and you don't get a good connection to the junction in the harness your car won't work either. If you are seeing burnt wires then I suggest repairing/replacing the harness to be sure of no hidden gremlins under the dash.
If the harness is damaged I suggest pulling it and rewiring it. You are gonna have to pull it to replace it, so if it doesn't look like you want to repair it after you get it out then at least you are halfway through the replacement job anyway. It's a day to two day job and not hard just time consuming because you want to check every wire in the harness just to be safe. It's a little easier in the newer cars where the entire dash frame can be pulled out completely. Some older cars can't be removed but I have rewired Challengers without pulling the dash. It wasn't as easy but it was doable.
As for why the Ammeter wires overheated.
1. All the current runs through the ammeter to get to the battery.
2. Old age corrodes connections and causes them to heat up. The ammeter being a high current connection it heats up more.
3. There is no current flow protection on the alternator side of the wiring so when you have a dead battery the alternator pumps out mega juice to fill it. Also if the voltage regulator/alternator malfunctions it can cause an overcharge problem. With old/bad connections only part of the current flows to the battery and the rest is lost in heat making the problem worse.
4. Eventually the heat takes it's toll on the fiber insulators under the ammeter and they degrade causing a dangerous potential for fire.
Imagine if you will an ammeter shorts from failed insulators while the car is running. First it pops the fusible link but with no overload protection on the alternator the car will still run pumping out juice into the shorted ammeter and poof.
If you repair the harness or buy a new one, put a fusible link on the alternator output. Cheap insurance in case of a short or bad voltage regulator that causes a runaway alternator problem. Also up-size the wires to the ammeter if you repair it yourself or augment the wires in the new harness for extra protection.
Also friends never let friends use crimp connectors only. Always solder, even if you crimp first.
Hope it helps.
